Abstract
Considered herein is the well-posedness, asymptotic stability and blow-up of the initial-boundary value problem for nonlocal singular viscoelastic wave equation with logarithmic nonlinearity \(u_{tt}-\frac{1}{x}(x u_{x})_x-\frac{1}{x}(x u_{xt})_x+\int \limits _{0}^{t}m(t-\lambda )\frac{1}{x}(x u_{x}(x, \lambda ))_x \hbox {d}\lambda =|u|^{r-2}u\ln |u|\) subject to a nonlocal boundary condition. Through the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differential-integral inequality techniques, we firstly demonstrate the global existence and uniqueness of weak solutions in certain weighted Sobolev spaces; Secondly, we establish the explicit polynomial and exponential energy decay estimates under some suitable conditions; Finally, we investigate the finite time blow-up criterion and then derive its upper and lower bounds of blow-up time. The above conclusions extend and improve some results in the literatures.
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1 Introduction
In the domain \(Q=[0,\iota ]\times [0,T]\) (\(0<\iota <\infty \)), we shall consider the following nonlocal singular viscoelastic problem with logarithmic nonlinearity term
where the exponent \(r>2\), and relaxation function \(m:[0,\infty ) \rightarrow (0,\infty )\) satisfies some special conditions to be given later. This type problem (1.1) is a classical nonlocal singular viscoelastic wave equation with a nonlocal (integral) condition \(\int _{0}^{l}\xi (x)u(x,t)\hbox {d}x=\gamma (x)\), which often appears in the heat conduction [1,2,3,4], electrochemistry [5], thermoelasticity [6], underground water flow [7] and other physical process. In general, the nonlocal (integral) boundary conditions arise in some mathematical models, where the exact boundary data cannot be measured directly, however their average value can be given exactly. For example, in some cases, the classical standard conditions, such as Dirichlet, Neumann and Robin type, are not always adequate to describe the pressure, temperature,... pointwise. But the nonlocal (integral) boundary conditions (average value of the solution) can be exactly measured along the boundary or some part of it. It is known that, from the physical point of view, viscoelastic materials possess a memory effect preserving their past traces and can be used to describe natural damping. In mathematics, these damping effect can be modeled by applying some integro-differential terms. For example, the formula \(\int \limits _{0}^{t}m(t-\lambda )\frac{1}{x}(x u_{x}(x, \lambda ))_x \hbox {d} \lambda \) in (1.1) is a typical viscoelastic term, and more information can be seen in [8, 9, 16, 17] and papers cited therein. Until now, much effort have be paid to study the qualitative properties of all kinds of viscoelastic parabolic and wave equations with classical and nonlocal boundary conditions. Not just the local (global) existence and uniqueness were established, but other properties (asymptotic behavior and regularity, etc.) were discussed as well.
1.1. Viscoelastic problem with classical boundary conditions
For the linear and nonlinear viscoelastic wave equations with classical standard boundary conditions, there exists an abounding results involving their initial-boundary value problem. For instance, Di and Song [8] considered the following quasilinear viscoelastic equation
subject to Dirichlet boundary condition. They established the global existence and nonexistence of weak solutions under low and critical initial energy by using Galerkin approximation method, improved potential well theory and some differential inequality techniques. When \(\rho =1\) and \(\alpha =\beta =0\) in (1.2), Berrimi and Messaoudi [9] established the local and global existence, and then gave a energy decay at enough small initial data; Kim and Han [10] derived the finite time blow-up at negative initial energy under suitable conditions on m. If \(\rho =1\) and \(\beta =0\) in (1.2), Song and Zhong [11] gave a finite time blow-up criteria under certain positive initial energy. Furthermore, Song and Xue [12] improved above conclusion of [11] to the solutions with arbitrary high initial energy. For \(\rho \ge 1\) and \(\alpha =0\) in (1.2), Messaoudi and Tatar [13] derived the global existence and uniform decay estimates by using potential well theory; Liu [14] also discussed the energy decay estimates by applying Lyapunov function technique and perturbed energy method. When the power nonlinear source \(|u|^{r-2}u\) in (1.2) is missing, Cavalcanti et al. [15] obtained the global existence for solutions at \(\alpha \ge 0\) and the exponential decay estimate for energy at \(\alpha >0\).
In [16], Xu, Yang and Liu studied the following viscoelastic wave equation
with Dirichlet boundary condition. The global existence was investigated at low initial energy; the finite time blow-up criteria were derived under positive initial energy.
Cavalcanti et al. [17] studied the variable coefficient viscoelastic wave equation
subject to Dirichlet boundary condition. Under certain conditions on m and \(a(x)+b(x)\ge \rho >0\) (\(\forall \) \(x\in \Omega \)), they gave that when m is decaying exponentially and h is linear, the energy is also exponential decay, and when m is decaying polynomially and h is nonlinear, the energy is decaying polynomially. When \(a(x)=1\), \(h(u_{t})=u_{t}\) and \(f(u)=|u|^{r}u\) in (1.4), Cavalcanti et al. [18] gave an exponential energy decay rate under some geometry restrictions on \(\omega \subset \Omega \) and assumptions on b(x) and m. Moreover, we also note that the viscoelastic wave equations with Neumann, Robin and classical standard mixed type boundary conditions have been attracted many authors’ attention, and several results involving their qualitative properties have been established (see [19,20,21,22] and references therein).
1.2. Wave equation with logarithmic nonlinearity
The logarithmic nonlinearity is an important and interest mathematical model, which appears in the super-symmetric physics, inflationary cosmology, quantum mechanics and nuclear fields [23,24,25]. Especially, the wave equations with logarithmic nonlinearity has be applied in various areas of science, such as viscoelastic theory, quantum mechanics etc (see [25, 26] and references therein). In the past thirty years, qualitative research on this type of equation has attracted widespread attention and achieved many excellent results; For example, in case of without presence viscoelastic term, Di, Shang and Song [27] investigated the following strongly damped wave equations with logarithmic nonlinearity
where if \(n=1,2\), \(2<r<+\infty \); if \(n\ge 3\), \(2<r<2^{*}=\frac{2n}{n-2}\). They first studied the global existence, uniqueness, exponential and polynomial energy decay estimates. Furthermore, they discussed the finite time blow-up criterion and got blow-up time estimation involving lower and upper bounds. When \(r=2\) in (1.5), Ma and Fang [28] investigated the global existence and energy decay estimates of weak solutions by applying potential well theory. In [29], Han investigated the initial-boundary value problem for the equation
where the global existence of weak solution was established by adopting Galerkin method, compactness principle and logarithmic Sobolev inequality. In the case of missing source term \(|u|^{2}u\) in (1.6), Ye [30] studied the above equation under the same initial and boundary conditions as [29], and discussed the global existence, blow-up and exponential decay estimations. Lian and Xu [31] studied the wave equation including the weak damping, strong damping and logarithmic source terms. The local existence, global existence, energy decay estimates and infinite time blow-up were discussed at different initial energy levels.
In [32], Hao and Du studied the initial-boundary value problem to viscoelastic wave equation
where the differential operators \(\mathcal {A}{u}=\text {div} (a_{ij}(x)\nabla u)\) with \(a_{ij}(x)\in C^{\infty }(\Omega )\). They analyzed the global existence, general decay rate and blow-up by using Riemannian geometry, logarithmic Sobolev inequality, Lyapunov functional method and potential well theory, etc. Liao [33] considered the following viscoelastic wave equation
and derived the finite time blow-up results and obtained the lifespan of the weak solution in high initial energy. Pikin, Agarwal and Irkl [34] proved the global existence and energy decay rate for the system of viscoelastic Kirchhoff type wave equations including logarithmic nonlinearity. More results for the wave equations involving viscoelastic term and logarithmic nonlinearity, we refer the reader to see [35, 36] and papers cited therein.
1.3. Nonlocal singular problem with nonlocal conditions
It is known that the nonlocal singular problem with nonlocal conditions have attracted a lot of attention for quite a long time, because they are very important and popular natural models appearing in all kinds of scientific fields. In mathematics, it has been studied by many scholars and also many outstanding results about the qualitative theory were emerged in recent years. For example, Mesloub and Bouziani [37] studied the singular wave equation
subject to nonlocal boundary conditions. They gave the blow-up result at large initial data, and derived the decay estimates at small initial data. Without the weak damping term \(u_{t}\) in (1.9), Mesloub and Bouziani [38] considered it with nonlocal condition and then gave the proof for the existence and uniqueness of strong solution.
In [39], Liu, Sun and Li studied the nonlocal singular viscoelastic wave equation
with nonlocal (integral) boundary condition \(\int \limits _{0}^{1}xu(x,t)\hbox {d}x=0\), where \(a\ge 0\). They used the potential well theory to achieve the blow-up results and energy decay estimation. When \(a=0\) in (1.10), Mesloub and Messaoudi [40] solved the global existence and energy decay, and they also gave the blow-up result at negative initial energy. Later, Wu [41] improved the results of [40] by giving the blow-up phenomenon at non-positive and positive initial energy levels. When the source term \(|u|^{r-2}u\) is replaced by general nonlinear term f(x, t, u), Mecheri et al. [42] gave the proof for the existence and uniqueness of the strong solutions. For more results in this research aspects, we provide the reader to see [36] and references cited therein.
In view of the above literatures, we first note that there is little conclusions on the qualitative properties of initial-boundary value problem for nonlocal singular viscoelastic wave equation including logarithmic nonlinearity subjected to a nonlocal boundary condition. Moreover, referring to the discussion in [28,29,30,31,32], we find that the classical logarithmic Sobolev inequality plays a key role in the study of parabolic and wave problems with logarithmic nonlinearity subjected to the classical standard boundary conditions; From the references [39,40,41,42], we see that the nonlocal singular wave problems (without logarithmic nonlinearity) are often analyzed in the weighted Sobolev spaces. However, the simultaneity appearance of nonlocal singular viscoelastic term, logarithmic nonlinear source term and nonlocal (integral) boundary condition in (1.1) cause some difficulties such that we cannot use the classical logarithmic Sobolev inequality and other some standard Sobolev type inequalities and techniques, when we study the qualitative theory of problem (1.1). At the same time, the interaction among above terms in (1.1) make that it requires a rather exquisite analysis in mathematical studies. Especially, we would like to know what will be happened to the well-posedness, blow-up and energy decay, etc. of the solutions for problem (1.1) ? It means that, compared with references [33, 34, 39,40,41], whether the appearing of nonlocal singular viscoelastic term, logarithmic nonlinear source term and nonlocal (integral) boundary condition will make some different effects on the qualitative properties. This question is a very interesting and opening. In fact, our results of this paper are quite different from cases: (1) viscoelastic wave equation with polynomial nonlinear term or logarithmic nonlinearity subject to classical boundary conditions; (2) nonlocal singular viscoelastic wave equations with polynomial nonlinear term and nonlocal boundary condition. Here, in the present paper, we shall use the effective combination of Galerkin approximation, modified potential well method, perturbed energy methods, convexity theory and differential–integral inequality techniques, and some new skills to study the well-posedness, blow-up and energy decay in certain weighted Sobolev spaces.
The plan of the rest of this article is as follows. In Sect. 2, we recall some weighted Sobolev spaces, potential well families, important lemmas and then state main results of this paper. In Sects. 3 and 4, we establish the global well-posedness and polynomial (and exponential) energy decay estimates of the solution. In the final section, we study the finite time blow-up criterion and derive blow-up time estimation involving upper and lower bounds.
2 Preliminaries and main results
In this section, we recall some basic concepts about the weighted Sobolev spaces and inequalities, and introduce some functional spaces and important lemmas associated with potential well theory. After that, we shall state the main conclusions of this paper.
Throughout the whole article, we denote the weighted \(L^{r}\)-space by \(L_{x}^{r}=L_{x}^{r}((0,\iota ))\) having the finite norm
When \(r=2\), \(L_x^2=L_{x}^{2}((0,\iota ))\) is the weighted Hilbert space of square integrable and its norm is \(\Vert u\Vert _{2}=\left( \int \limits _{0}^{\iota }x|u|^{2}\hbox {d}x\right) ^\frac{1}{2}\). The corresponding scalar product is \((u,v)_{L_{x}^{2}}=(xu,v)\). We define \(H=H_{x}^{1,2}((0,\iota ))\) to be the weighted Hilbert space with the norm
and further introduce the functional spaces
Lemma 2.1
([39, 40], Poincaré-type inequality) \(\forall \) \( u\in H_{0}\), we have
where B is a certain positive constant.
Lemma 2.2
[39, 40] \(\forall \) \( u\in H_{0}\) and \(2<r<4\), we have
where \(B_{s}\) is the normal number that depends only on \(\iota \) and r.
Remark 2.1
Here, it is clear that we can define an equivalent norm \(\Vert u\Vert _{H_{0}}=\Vert u_{x}\Vert _{2}\) for \(u\in H_{0}\).
For any \((u,u_{t})\in V_{0}\), we introduce some functionals and potential well sets as follows:
where N is the Nehari manifold, d denote the depth of potential well, and the function
By the definitions of J(u) and I(u), we have
In addition, we will introduce the following set:
From the above sets and the definition of d, we know that \(M^{+}\cap M^{-}=\emptyset \), \(M^{+}\cup M^{-}=M\).
Now, we will give some basic hypotheses on the relaxation function m to obtain the main conclusions of this paper.
-
(A1)
\(m:[0,+\infty )\rightarrow R^{+}\) is a non-increasing \(C^1\) function, i.e., \(m(0)>0\), \(m^{\prime }(\lambda ) \le 0\), and satisfies
$$\begin{aligned} k(t)>1-\int \limits _{0}^{\infty }m(\lambda )\hbox {d}\lambda =k>0. \end{aligned}$$(2.7) -
(A2)
There exists a positive constant \(\delta \) satisfying that for \(t\ge 0\),
$$\begin{aligned} m^{\prime }(t)\le -{\delta }m^{p}(t),\quad 1 \le p<\frac{3}{2}. \end{aligned}$$(2.8)
Remark 2.2
Noting that form the condition \(1 \le p<\frac{3}{2}\), we can deduce \(\int \limits _{0}^{\infty }m^{2-p}(\lambda )\hbox {d}\lambda <\infty .\)
In fact, if \(p=1\), the assumption (A2) means that \(m(\lambda )\le m(0) e^{-\delta \lambda }.\) This immediately implies that \(\int \limits _0^{\infty }m^{2-p}(\lambda )\hbox {d}\lambda <\infty .\)
When \(1<p<3/2\), by the assumption (A2), we obtain that \(m(\lambda )\le \frac{m(0)}{(1+a\lambda )^\frac{1}{p-1}}.\) Since \(\frac{2-p}{p-1}>1\), it follows that
where \(a=\delta (p-1)/m^{p-1}(0)\).
We consider the weak solutions to problem (1.1) as defined below.
Definition 2.1
The function u is called the weak solutions to problem (1.1) on \((0,\iota )\times (0,T)\), if \(u\in {L^{\infty }(0,T;H_{0})}\), \(u_{t} \in {L^{\infty }(0,T;L_{x}^{2})}\cap {L^{2}(0,T;H_{0})}\) and satisfies
-
(1)
\(\forall \) \(v \in H_{0}\), a.e. \(t\in [0,T]\),
$$\begin{aligned} \begin{aligned}&(xu_{t},v)+\int \limits _{0}^{t}\left( xu_{x},v_{x}\right) \hbox {d}\lambda +\left( xu_{x},v_{x}\right) =\int \limits _{0}^{t}(x|u|^{r-2}u\ln |u|,v)\hbox {d}\lambda \\&\quad +\int \limits _{0}^{t}\left( x\int \limits _{0}^{\lambda }m(\lambda -\tau )u_{x}(\tau )\hbox {d}\tau ,v_{x}\right) \hbox {d}\lambda +(xu_{1},v)+(xu_{x0},v_{x}); \end{aligned} \end{aligned}$$(2.9) -
(2)
\(u(x,0)=u_{0}\) in \(H_{0}\), \(u_{t}(x,0)= u_{1}\) in \(L_{x}^2\).
Here, replacing v by \(u_{t}\) in (2.9) and a direct calculations give the following lemma.
Lemma 2.3
(Energy inequality) Let u be the weak solutions to problem (1.1). Then we have that E(t) is a non-increasing function with respect to time t, and satisfies the following energy equality for all \(t\in {[0,\infty )}\),
The following lemmas states some basic properties involving with potential well theory. Details of the proof are given in Appendix A.
Lemma 2.4
Suppose that \(u\in H_{0}\backslash \{0\} \), then it follows that
-
(1)
\(\mathop {\lim }\limits _{\omega \rightarrow 0^+}J(\omega u)=0\), \(\mathop {\lim }\limits _{\omega \rightarrow +\infty }J(\omega u)={-\infty };\)
-
(2)
There exists a unique number \(\omega _*>0\) satisfying that \(\frac{\textrm{d}}{\textrm{d}\omega }J(\omega u)|_{\omega =\omega _*}=0;\)
-
(3)
\(J(\omega u) \) is an increasing function on \(0<\omega <\omega _*\), a decreasing function on \(\omega _*<\omega <{+\infty }\), and attains its maximum value at \(\omega =\omega _*;\)
-
(4)
\(I(\omega u)>0 \) on \(0<\omega <\omega _*\), \(I(\omega u)<0 \) on \(\omega _*<\omega <{+\infty }\), and \(I(\omega _*u)=0\).
Remark 2.3
From part (4) in Lemma 2.4, we can see that N is not empty. Thus, the definition (2.5) of potential well depth d is meaningful.
Lemma 2.5
The potential well depth d defined in (2.5) is a positive number; And there exists a positive function \(u \in N \) satisfying \(J(u)=d.\)
Now, we shall in a position to give major theorems of this article, and their proofs will be given in subsequent sections.
Theorem 2.1
(Global well-posedness) Suppose that (A1) holds, \(2<r<3\) and \((u_0,u_1)\in M^+\). Then the problem (1.1) have a unique global weak solution \(u\in L^{\infty }(0,\infty ;H_{0})\), \(u_t\in L^{2}(0,\infty ;H_{0})\cap L^{\infty }(0,\infty ;L_{x}^{2})\) and \((u,u_{t})\in M^+\) for all \(t\ge 0\).
Theorem 2.2
(Energy decay estimates) Suppose that (A1), (A2) hold, \(2<r<3\) and \((u_0,u_1)\in M^+\). Let u to be a global weak solution of problem (1.1). Then there exist three positive constants \(L_{1},L_{2}\) and \(\zeta \) satisfying
Remark 2.4
Note that the energy decay estimates in [39, 40] hold for \(t\ge t_{0}>0\). However, by applying some new skills, we improved their results and conclude that (2.11) involving energy decay are holding for all \(t\ge 0\).
Theorem 2.3
(Blow-up I) Suppose that (A1) holds, \(2<r<\infty \) and \((u_{0},u_{1})\in M^-\). Further assume that \(E(0)<\varrho d\) (\(\varrho <1\)) and
where \(\hat{\varrho }=max\{0, \varrho \}\). Then the solutions u of problem (1.1) blow-up in finite time, which means that there exists the maximum existence finite time \(T_*\) such that
In addition, the upper bound of above blow-up time \(T_*\) is given by
where b and \(T_{0}\) are positive numbers depending on \(u_{0}\), \(u_{1}\), r and determining later.
Theorem 2.4
(Blow-up II) Under the conditions of Theorem 2.3 and \(2<r<3\), then the solutions u of problem (1.1) is unbounded in finite time \(T_*\) and satisfies
In addition, the lower bound of above blow-up time \(T_*\) is given in form
where \(0<\nu <3-r\), \(B_1\), \(B_6\) are the optimal Sobolev constants satisfying \(\Vert u\Vert _{r+\nu }\le B_1\Vert u\Vert _{H_{0}}\), \(\Vert u\Vert _{2(r-1+\nu )}\le B_6\Vert u\Vert _{H_{0}}\), and \(Q(0)=\Vert u_1\Vert _{2}^{2}+\Vert u_{0}\Vert _{H_{0}}^{2}.\)
Remark 2.5
Here, the blow-up result (2.13) appearing in Theorem 2.4 implies that the conclusion of blow-up (2.14) also holds.
In view of Theorems 2.1 and 2.3, we conclude a threshold result of global existence and nonexistence as follows.
Corollary 2.1
Let \(2<r<3 \) and \(E(0)<\varrho d\) \((\varrho <1)\). Then we have that problem (1.1) has a global weak solution provided \(I(u_{0})\ge 0\), and does not exists any global weak solution provided \(I(u_{0})<0\).
3 Global existence and uniqueness
The aim of this section is focus on the proof of global existence and uniqueness of weak solutions to problem (1.1) provided that the initial data come from \( M_{1}^{+}\) and \(M_{2}^{+}\).
Proof involving Theorem 2.1 is divided into 4 steps.
Proof
Step 1. Global existence for the case \((u_{0},u_{1})\in M_{1}^{+}\) Since \((u_{0},u_{1})\in M_{1}^{+}\) i.e., \(E(0)<d \) and \(I(u_{0})\ge 0,\) then we shall consider it as follows.
-
(1)
\(E(0)<0 \) and \(I(u_{0})\ge 0\); It is contradictive with (2.3) and (2.6).
-
(2)
\(E(0)=0 \) and \(I(u_{0})\ge 0\); By (2.1), (2.3) and (2.6), it is easy to get \((u_{0},u_{1})\equiv (0,0),\) which is a trivial case.
-
(3)
\(0<E(0)<d\) and \(I(u_{0})=0\); This is impossible and contradicts with the definition of d.
-
(4)
\(0<E(0)<d\) and \(I(u_{0})>0\) is the only case we need to consider here.
Thus, according to Galerkin’s method, we first choose \(\{\varphi _{j}\}_{j=1}^{\infty }\) to be orthogonal basis of \(H_{0}\), and then construct approximation solutions \(u^{(n)}\) for problem (1.1) by
which satisfies the following initial value problem
Taking derivative of (3.2) with respect to t, and then multiplying it by \(c_{jt}^{n}\) and summing for \(j=1,2,......\) gives that
Via a series of calculation, it follows that
and
Inserting (3.6)–(3.8) into (3.5) and integrating it over (0, t), then we discover
Applying (3.3), (3.4) and the continuity of \(E(u,u_{t})\) and I(u), it is found that for n large enough and \(0\le t<\infty ,\)
which implies that \((u^{(n)}_{0},u^{(n)}_{1})\in M_{1}^{+}\).
Furthermore, we shall prove that \((u^{(n)},u^{(n)}_{t})\in M_{1}^{+}\) for n large enough and \(0\le t<\infty .\) In fact, if this is not true, then there exist a \(t_{0}\in (0,\infty )\) such that \((u^{(n)}(t_{0}),u^{(n)}_{t}(t_{0}) \in \partial M_{1}^{+}\), i.e.,
Thus, by (3.9) and (3.10), it is easy to see that \(E(u^{(n)}(t_{0}),u^{(n)}_{t}(t_{0}))=d\) is not true. Moreover, according to the definition of d and \(I(u^{(n)}(t_{0}))=0\), we can obatin that \(E(u^{(n)}(t_{0}),u^{(n)}_{t}(t_{0}))>J(u^{(n)}(t_{0}))\ge d\), which is also contradictive with (3.9) and (3.10). Thus, for n enough large and \(0\le t<\infty \), we have that \((u^{(n)},u^{(n)}_{t})\in M_1^+\) and also satisfies
So we have
Therefore, the estimates (3.11) and (3.12) allow us to obtain a subsequence of \(\{u^{(n)}\}\) which will be still denote by \(\{u^{(n)}\}\) satisfying
Since \(H_{0}\hookrightarrow L_{x}^{2}\) is compactness, we have form Aubin–Lions theorem that
which implies that
Immediately, combine the Schwarz’s inequality with (3.11) and (3.12) gives that
Furthermore, we choose \(r' =\frac{r}{r-1}\) and \(\nu >0\) small enough such that \((r-1+\nu )r'<4\). Therefore, by a direct calculation and weighted Sobolev inequality, there appears the relation
Here, we used the weighted Sobolev embedding inequality \(\Vert u^{(n)}\Vert _{ (r-1+\nu )r'} \le B_2\Vert u^{(n)}\Vert _{H_{0}}\), and applied \(|y^{r-1}\ln y|\le (e(r-1))^{-1}\) (\(0<y<1\)), while \(y^{-\nu }\ln y\le (e\nu )^{-1}\) (\( y\ge 1,\nu >0\)). Hence, from (3.19), (3.23) and Lions Lemma (see [26], Lemma 1.3), it follows that
Taking \( n\rightarrow \infty \) in (3.3) and (3.4), we see that \(u(x,0)=u_{0}\) in \( H_{0} \), and \(u_t(x,0)=u_1\) in \(L_{x}^{2}\). Furthermore, passing to the limit in (3.2), it is found that for \(\forall \) \(\varphi \in H_{0}\) and \(t\ge 0\),
Hence, we deduce that u is a global weak solution to problem (1.1). Lastly, from \((u_{0},u_{1})\in M_{1}^{+}\) and a standard contradiction argument as previous step, we can conclude that \((u,u_{t})\in M_{1}^{+}\) for all \(t\ge 0\). Hence, the proofs are finished.
Step 2. Global existence for the case \((u_{0},u_{1})\in M_{2}^{+}\).
We will achieve this step by considering two aspects \(I(u_{0})>0\) and \(I(u_{0})=0\).
(i) When \(I(u_{0})>0\), a direct calculation gives that \(\omega \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u_{0})|_{\omega =1}=I(\omega u_{0})|_{\omega =1}>0\). So there exists an interval \((\omega _\alpha , \omega _\beta )\) such that \(1\in (\omega _\alpha ,\omega _\beta ) \) and \(\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u_{0})>0\), \(I(\omega u_{0})>0\) for \(\omega \in (\omega _\alpha ,\omega _\beta ).\) In doing so, we can take a sequence \(\{\omega _n\}_{n=1}^{\infty }\) such that \(\omega _\alpha<\omega _n<1\) and \(\lim \limits _{n\rightarrow \infty }\omega _{n}=1\).
Thus, let us choose \(u^{(n)}_{0}=\omega _nu_{0}\) and \(u^{(n)}_{1}=\omega _nu_{1}\), and study the problem (1.1) with initial conditions
where the previous initial data \(u_{0}\), \(u_{1}\) in (1.1) are replaced by \(u_{0}^{(n)},u^{(n)}_{1}.\)
Since \(I(u_{0})>0\), by lemma 2.4 (4) we have that \(\omega _*=\omega _*(u_0)>1.\) Therefore, the combination of \(0<\omega _n<1\) and \(J(u^{(n)}_{0})=\frac{r-2}{2r}\Vert u^{(n)}_{0}\Vert _{H_{0}}^{2}+\frac{1}{r}I(u^{(n)}_{0})+\frac{1}{r^2}\Vert u^{(n)}_{0}\Vert _r^r\), there appears the relations
and for sufficiently large n,
(ii) If \(I(u_{0})=0\), the definition of d gives that \(J(u_{0})\ge d\). However, from \(u_{1}\in L_{x}^{2}\backslash \{0\} \) and
we have \(J(u_{0})<d\). So there exists a contradiction.
Based on above discussion, we obtain \((u^{(n)}_{0},u^{(n)}_{1})\in M_{1}^{+}\). The similar proof as step 1, for every n we have that problem (1.1) (initial condition \(u_{0}\), \(u_{1} \) are replaced by \(u^{(n)}_{0},u^{(n)}_{1}\)) has a global weak solution \(u^{(n)}\in L^\infty (0,\infty ;H_{0})\), \(u_{t}^{(n)}\in {L^{\infty }(0,T;L_{x}^{2})}\cap {L^{2}(0,T;H_{0})}\) and \((u^{n},u^{n}_{t})\in M_{1}^{+},\) which also satisfies for any \(v\in H_{0}\) and \(t\ge 0\),
Similarly, we can obtain the estimates (3.11), (3.12), (3.22) and (3.23) as the proof of step 1. Thus, there exists a subsequence still by \(\{u^{(n)}\}\) and \((u,u_t)\in M_2^+\) such that
Now, according to the definition of \(u_{0}^{(n)}\), \(u^{(n)}_{1}\) and (3.27), it follows that \(u(x,0)=u_{0} \) in \(H_{0}\), and \(u_t(x,0)=u_{1}\) in \(L_{x}^{2}.\) Furthermore, taking limit in (3.31), we get that for any \(v\in H_{0}\) and \(n\rightarrow \infty \),
Hence, we deduce that u is a global weak solution to problem (1.1) in case \((u_{0},u_{1})\in M_{2}^{+}\). Moreover, \((u,u_{t})\in M_{2}^{+}\) (\(t>0\)) can also be proved in the similar way as step 1 and details are omitted here.
Step 3. Uniqueness for the case \((u_{0},u_{1})\in M_{1}^{+}\).
Suppose that there exist two solutions \(u^{(a)}\) and \(u^{(b)}\) to problem (1.1). Thus, \(w=u^{(a)}-u^{(b)}\) solves the following problem
in domain \([0,\iota ]\times [0,\infty )\), where \(f(z)=|z|^{r-2}z\ln |z|\) for \(z\in \mathbb {R}.\)
Multiplying (3.32) by \(xw_t\), and then integrating it over (0, l), it follows that
where \(0<\rho <1\). By Hölder and Sobolev inequalities, we have
where we choose \(\kappa _{1}\in (\frac{2}{r-2},\frac{4}{r-2})\) such that \(2<\kappa _{1}(r-2)<4\) and \(2<\frac{2\kappa _{1}(r-2)+12}{\kappa _{1}(r-2)+4}<3\). Thus, \(B_{3}\) and \(B_{4}\) are the optimal constants satisfying \(\Vert u^{(a)}+\rho u^{(b)}\Vert _{\kappa _{1}(r-2)}\le B_3\Vert u^{(a)}_{x}+\rho u^{(b)}_{x}\Vert _{2}\) and \(\Vert w\Vert _{\frac{2\kappa _{1}(r-2)+12}{\kappa _{1}(r-2)+4}}\le B_4\Vert w\Vert _{H_{0}}\), respectively.
Furthermore, choosing \(\kappa _{2} \in (\frac{2}{r},\frac{4}{r})\) and \(\nu >0\) small enough such that \(2<(r-2+\nu )\varpi <4\), where \(\varpi =\frac{r\kappa _{2}}{r-2}\), and using the same calculation to (3.23), it follows that
where \(B_5\) is the embedding constant satisfying \(\Vert u^{(a)}+\rho u^{(b)}\Vert _{(r-2+\nu )\varpi }\le B_5\Vert u_x^{(a)}+\rho u_x^{(b)}\Vert _2\). Here, note that when \(2<r<3\), we have \(\frac{r-2}{r\kappa _{2}}+\frac{r\kappa _{2}-2r+4}{2r\kappa _{2}}+\frac{1}{2}=1\) and \(2<\frac{2r\kappa _{2}}{r\kappa _{2}-2r+4}<4\). So, using Hölder and Sobolev inequalities again, we discover the relation
Then, by a direct calculation, we obtain
By the combination of (3.33)–(3.37), assumption conditions (A2) and \(k<k(t)<1\), we obtain that
Thus, employing the Gronwall Lemma and initial data \(w(x,0)=0\), \(w_{t}(x,0)=0\), we conclude that \(\Vert w_{t}\Vert _{2}^{2}+k(t)\Vert w\Vert _{H_{0}}^{2}+(m\circ w_{x})(t)=0,\) which implies that \(w=0\) a.e. in \((0,\iota )\times [0,\infty ).\)
Step 4. Uniqueness for the case \((u_0,u_1)\in M_2^+\).
The proof of this case are alike to the steps 3. This completes the proof of Theorem 2.1. \(\square \)
4 Energy decay estimates
In this section, we shall turn to prove the exponential and polynomial energy decaying (Theorem 2.2) of the solutions to problem (1.1) when initial data \((u_{0},u_{1})\in M^{+}\).
To complete this process, one can follow the similar approach as [9, 39, 40, 44, 45] and introduce the functional
where the constants \(\varepsilon _1\), \(\varepsilon _2\) are positive and
Lemma 4.1
Let \(\varepsilon _1\) and \(\varepsilon _{2}\) be small enough. Then under conditions of Theorem 2.2, there exist two constants \(0<\mu _{1}<1\) and \(\mu _{2}>1\) such that
Proof
Since \( (u_{0},u_{1}) \in M^{+}\), form the conclusions of Theorem 2.1, we have obtained that \( (u,u_{t}) \in M^{+}\) for all \(t\ge 0\). Hence, by (2.3) and (2.6), we have
Thus, a straightforward computations give that
where the constant B was defined in Lemma 2.1. Along the similar way, it follows that
for \(\varepsilon _{1}\) and \(\varepsilon _{2}\) small enough and \(0<\mu _{1}<1\). \(\square \)
The following Lemmas 4.2 and 4.3 are similar to the lemmas of [40] with suitable modification, and details are then omitted.
Lemma 4.2
Suppose that \(0<\vartheta <1 \), \(p>1\), and (A1), (A2) hold. Let \(u\in L^{\infty }((0,T);H_{0})\) and m be the relaxation function. Then there exists a positive constant C such that
Lemma 4.3
Suppose that \(0<\vartheta <1 \), \(p>1\), and (A1), (A2) hold. Let \(u\in L^{\infty }((0,T);H_{0})\) and m be the relaxation function. Then there exists a positive constant C such that
Lemma 4.4
Let (A1), (A2) hold. Then for all \(\alpha >0\) the function \( \varphi \) defined in (4.2) satisfies
Proof
Utilizing the equation in (1.1), it follows that
By the Young inequality, we have
Note that \(1-k(t)=\int \limits _{0}^{t}m(\lambda )\hbox {d}\lambda <\int \limits _{0}^{\infty } m(\lambda )\hbox {d}\lambda =1-k\) and using Young inequality again, we obtain
In view of (4.9)–(4.11), we have
Taking \(\gamma =k/(1-k),\) then we conclude that (4.8) is holding. \(\square \)
Lemma 4.5
Assume \(2<r<3\), \(0<\nu <3-r\) and (A1), (A2) hold. Then for all \(\beta >0\) the function \( \psi \) defined in (4.3) satisfies
where \(B_{6}\) is the optimal Sobolev constant satisfying \(\Vert u\Vert _{2(r-1+\nu )} \le B_{6}\Vert u\Vert _{H_{0}}\); B and \(B_{s}\) are defined in Lemmas 2.1 and 2.2, respectively.
Proof
By the equation in (1.1), a direct calculations give that
Next, we shall estimate the right hand side of (4.14). For \(\beta >0\), using Young and Hölder inequalities, it follows that
The same calculation gives that
Let \(0<\nu <3-r\) such that \(2<2(r-1+\nu )<4\). Applying the inequality \(y^{-\nu }\ln y\le (e\nu )^{-1}\) (\( y>0,\nu >0\)), Young inequality and Lemma 2.1, we have
For the fourth and fifth terms, a series of calculations give that
and
The similar derivative as (3.11), we have \(\Vert u\Vert _{H_{0}}^{2}\le \frac{2rd}{(r-2)k}\). Thus, the sixth term can be estimated by
Hence, in view of (4.14)–(4.19), we conclude that (4.13) holds. \(\square \)
Based on above lemmas, we will in a positive to give the proof for Theorem 2.2.
Proof
By the combination of Lemmas 2.3, 4.4 and 4.5, we have
where B was introduced in Lemma 2.1. Here, we first pick \(\alpha \) small enough such that
Once \(\alpha \) is fixed, we can choose \(\varepsilon _{2}\) small enough such that
which shows that
Furthermore, we choose \(\varepsilon _{1},\varepsilon _2\) so small that (4.4) and (4.21) are valid and
Hence, (4.20) becomes
Next, we shall consider the following two cases for the different range of p.
Case (I). When \(p=1\), combining (4.4) and (4.22) gives that
for some \(\eta _{1}>0\). Integration (4.23) over 0 to t leads to
Applying (4.4) again, we have
Choosing \(L_{1}= \frac{1}{\mu _{1}}D(0)\) and \(\zeta =\frac{\eta _{1}}{\mu _{2}}\), it follows that \(E(t)\le L_{1}e^{-\zeta t}, \quad \forall \ t\ge 0.\)
Case (II). If \(1<p<\frac{3}{2}\), we obtain from (A2) that
A direct calculation fives that for all \(0<\vartheta <1\),
Let \(0<\vartheta<2-p<1\). We have \(\frac{1-\vartheta }{p-1}>1.\) Hence, we deduce
Similar way as (3.11), we have \(\Vert u\Vert _{H_{0}}^{2}\le \frac{2rd}{(r-2)k}\). By Lemmas 2.1 and 4.2, we discover the relation
for some positive constant C. Hence, for any \(r_{1}>1\), we have
Here, we can choose \(\vartheta =1/2\) and \(r_{1}=2p-1\), that means \({\vartheta r_{1}/(p-1+\vartheta )}=1 \). Thus, we have from (4.28) that for some \(\xi >0,\)
In view of (4.4), (4.22), (4.29), it follows that
A direct integration of (4.30) yields to
Hence,
Since \(1/(r_{1}-1)>1\) and \(1+t\rightarrow +\infty \) (\(t\rightarrow +\infty ),\) we get
Because E is bounded, so we use (4.4) and (4.33) to get
Then, applying Lemmas 4.1 and 4.3, we discover
which implies
for some positive constant C. Therefore, combining (4.22) and (4.36), we get
On the other hand, the same arguments as (4.29) gives that
Combining (4.37) and (4.38), we discover that
Integrating (4.39) from 0 to t, we have
In view of (4.4) with (4.40), we deduce \( E(t)\le L_{2}(1+t)^{-1/(p-1)}, \ \forall \ t\ge 0.\) Consequently, we finish the proof of Theorem 2.2. \(\square \)
5 Finite time blow-up
In this section, we shall pay attention to the proof of finite time blow-up for initial data \((u_{0},u_{1})\in M^{-}\); And we also give the estimates of upper and lower bounds of blow-up time to problem (1.1).
We first introduce two lemmas which will pay an significant role to obtain main results.
Lemma 5.1
Let \((u_{0},u_{1})\in M^{-}\) and \(2<r<\infty \). Then, for all \(t\ge 0\) we have \((u,u_{t})\in M^{-}\).
Proof
By applying contradiction, if this is not true, then we suppose that \((u,u_{t})\) leaves \( M^{-}\) at time \(t=t_{0}\). Thus, we know that \(I(u(t_{0}))=0\) or \(E(u(t_{0}))>d.\) And, we can find a time sequence \(\{t_{n}\}\) (\(t_{n}\rightarrow t_{0}\)) satisfying \(I(u(t_{n})<0\) and \(E(t_{n})\le d.\) However, utilizing the weak lower semi-continuity of \(\Vert \cdot \Vert _{H_{0}}\), we obtain that
and
So \(E(u(t_{0}))>d\) is impossible, which contradicts (5.2). Once \(I(u(t_{0}))=0\), from the definitions of d and E(t), it follows that
which brings to a contradiction. We thereby get that \((u,u_{t})\in M^{-}\) for all \(t\ge 0.\) \(\square \)
Lemma 5.2
[43] Let K(t) be a positive \(C^2\) function and satisfy the inequality
for \(\varsigma >0 \) and \(t>0\). Furthermore, when initial data \(K(0)>0\) and \(K^{\prime }(0)>0,\) there exists the time \(T_{*}\le {\frac{K(0)}{\varsigma K^{\prime }(0)}}\) such that
5.1 Upper bound for blow-up time
This subsection is devoted to prove the finite time blow-up criterion, and estimate the upper bound of blow-up time (Theorem 2.3).
Proof
Arguments by contradiction, we fist assume that the solution u is global existence. Thus, for any \(t>0\), we can define function K : \([0,T]\rightarrow R^+\) by
where the positive constants b and \(T_0\) will be determined later. A series of direct calculation gives that
and
Multiplying equation in (1.1) with xu and then adding the result into \(K^{\prime \prime }(t)\), it follows that
for \(t\in [0,T]\). Thus, there shows up the relation
where
Furthermore, by Schwarz’s inequality, we have that \(H(t)\ge 0 \) for any \(t\in [0,T].\) Hence, we derive the following relation
where \(M(t):[0,T]\rightarrow \mathbb {R}\) is defined in form
By Lemma 2.3, we have
for all \(t\in [0,T]\). Applying Young inequality, for \(\epsilon >0\) we have
Substituting (5.7) into (5.6) yields that
Next, we shall consider two cases according to the range of \(\varrho \).
Case 1: When \(\varrho <0\), which implies \(E(0)<0\). We choose \(\epsilon =r\) in (5.8) and \(0<b\le -2E(0)\). Thus, we have from (2.12) that
Case 2: If \(0<\varrho <1\), i.e., \(E(0)<\varrho d<d\). Picking \(\epsilon =(1-\varrho )r+2\varrho \) and \(b=2(\varrho d-E(0))>0\) in (5.8), it follows that
Applying (2.12) again, we have the relation
Moreover, since \((u_{0},u_{1})\in M^{-}\), we have from Lemma 5.1 that \((u,u_{t})\in M^-\) for \(t\ge 0\). So we have \(I(u)<0\) for \(t\ge 0\). Hence, by Lemma 2.4 (4), we obtain that there exist a \(\omega _{*}\in (0,1)\) such that \(I(\omega _{*}u)=0.\) Therefore, combining (2.6) and definition of d yields that
By (5.9), (5.10) and (5.12), we have
Therefore, we deduce that \(M(t)\ge 0\) for cases 1 and 2, which together with (5.5) yields
By (5.4), we can see that \(K(0)=\Vert u_0\Vert _2^2+T\Vert u_{0}\Vert _{H_{0}}^{2}+bT_{0}^{2} >0.\) In doing so, we take \(T_{0}\) large enough such that
which implies that
Thus, using Lemma 5.2, we obtain that there exits a time
such that \(K(t)\rightarrow \infty \) as \(t\rightarrow T_*^{-}\).
Finally, for a fixed \(T_0\), we can choose T such that
Combing (5.17) and (5.18), we get \(T>T_*\), which contradicts to our assumption. Thus, we end the proof of this theorem. \(\square \)
5.2 Lower bound for blow-up time
Our gold of this subsection is to give the proof of lower bound estimate (Theorem 2.4) for blow-up time to problem (1.1).
Proof
We first introduce the function
Furthermore, differentiating Q(t) with respect to t, we have form (1.1) that
Then, we choose \(0<\nu <3-r\) such that \(2<2(r-1+\nu )<4\). Thus, by the Young and Sobolev inequalities, it follows that
where \(B_{2}\) is the embedding Sobolev constant satisfying \(\Vert u\Vert _{2(r-1+\nu )} \le B_{6}\Vert u\Vert _{H_{0}}.\) And we have used \(|y^{r-1}\ln y|\le (e(r-1))^{-1}\) (\(0<y<1\)), \(y^{-\nu }\ln y\le (e\nu )^{-1}\) (\( y\ge 1,\nu >0\)). By Young’s and Schwarz’s inequalities, we discover that
Applying Lemma 5.1, we have \(I(u)<0\), which combine with (2.2) gives
Inserting (5.23) into (5.22), and then using the fact \(y^{-\nu }\ln y\le (e\nu )^{-1}\) (\( y>0,\nu >0\)) yields
where the constant \(B_{1}\) satisfies Sobolev embedding inequality \(\Vert u\Vert _{p+\nu }\le B_{1}\Vert u\Vert _{H_{0}}\).
Therefore, in view of (5.20), (5.21) and (5.24), it follows that
Integrating above inequality (5.25) over (0, t), we get
From the conclusions of Theorem 2.3, we note that the solutions u blow up in finite time \(T_*\), i.e., \(\lim _{t\rightarrow T_*}Q(t)=+\infty .\) Hence, we derive a lower bound estimation for \(T_*\) satisfying
Clearly, the above integral is bounded because the exponent is \(1<r-1+\nu <2\). We thereby complete the proof of Theorem 2.4. \(\square \)
6 Conclusion and future work
The goal of this article is to study the well-posedness, asymptotic stability and blow-up of the solutions for initial-boundary value problem (1.1). It is worth mentioning that the simultaneity appearance of nonlocal singular viscoelastic term, logarithmic nonlinear source and nonlocal (integral) boundary condition in (1.1) bring some difficulties such that the classical logarithmic Sobolev inequality, other some standard Sobolev type inequalities and techniques cannot be used here, and the interaction among these terms also requires a rather exquisite analysis in mathematical aspects, when we study the qualitative properties of problem (1.1). Considering the above situations, in the weighted Sobolev spaces, by the effective combining of Galerkin approximation method, modified potential well theory, perturbed energy method, convexity theory and differential-integral inequality techniques, we analyze the global existence and uniqueness (Theorem 2.1), the polynomial and exponential energy decay rates (Theorem 2.2), the finite time blow-up criterion and its upper and lower bounds estimations (Theorems 2.3 and 2.4). Especially, we also derive a threshold result of global existence and nonexistence for the solutions (Corollary 2.1).
Compared with some previous works involving the viscoelastic wave equation with polynomial nonlinear term or logarithmic nonlinearity subject to classical boundary conditions (see [33, 34]) and nonlocal singular viscoelastic wave equations with polynomial nonlinear term and nonlocal boundary conditions (see [39,40,41]), we find that our conclusions of this paper improve and extend their earlier well-posedness, asymptotic stability and blow-up results etc. However, all conclusions of this paper are established at low and critical initial energy \(E(0)\le d\) and case \(r>2\). On the one hand, our conclusions do not hold for critical case \(r=2\) because of the weighted Poincaré-type inequality and restrictive condition (2.12) appear in the proof. So we need to find some new methods and techniques to solve above difficulties, and this is an interesting problem. On the other hand, as far as we know there is little information about the qualitative properties for high initial energy level \(E(0)>d\). More precisely, at high initial energy level, we have not fully exploited to see that if problem (1.1) has the global existence and asymptotic property, and whether there exists finite time blow-up at some certain conditions. These questions are all opening, and we are now working on these problems.
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Appendix A
Appendix A
For the readers’ convenience, we will provide the details about the proofs of Lemmas 2.4 and 2.5 in this Appendix.
Proof
(Proof of Lemma 2.4.)
-
(1)
Applying the definition of J(u), we discover
$$\begin{aligned} J(\omega u)=\frac{\omega ^{2}}{2}\big (k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)\big )+\frac{\omega ^{r}}{r^{2}}\big (1-r\ln \omega \big )\Vert u\Vert _{r}^{r}-\frac{\omega ^{r}}{r}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|\hbox {d}x. \end{aligned}$$(6.1)Hence, considering \(\Vert u\Vert _{r}\ne 0\), we can see that (1) holds;
-
(2)
By a series of calculations, we have
$$\begin{aligned} \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u) =\omega \left( k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})\left( t\right) -\omega ^{r-2}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x-\omega ^{r-2}\ln \omega \Vert u\Vert _{r}^{r}\right) . \end{aligned}$$(6.2)Let \(\omega f(\omega )=\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)\), where
$$\begin{aligned} f(\omega ):=k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)-\omega ^{r-2}\int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x-\omega ^{r-2}\ln \omega \Vert u\Vert _{r}^{r}. \end{aligned}$$So we can show that
$$\begin{aligned} f^{\prime }(\omega )= -\omega ^{r-3}\left[ \left( r-2\right) \int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x+\left( r-2\right) \ln \omega \Vert u\Vert _{r}^{r}+\Vert u\Vert _{r}^{r}\right] . \end{aligned}$$If we choose
$$\begin{aligned} \omega _{1}=\exp \left( \frac{\left( r-2\right) \int \limits _{0}^{\iota }x|u|^{r}\ln |u|d x+\Vert u\Vert _{r}^{r}}{\left( 2-r\right) \Vert u\Vert _{r}^{r}}\right) >0, \end{aligned}$$(6.3)it follows that \(f^{\prime }(\omega _{1})=0\), \(f^{\prime }(\omega )>0\), for \(0<\omega <\omega _{1}\) and \(f^{\prime }(\omega )<0 \) for \(\omega >\omega _{1}\). Since
$$\begin{aligned} \lim _{\omega \rightarrow 0}f\left( w\right) =k(t)\Vert u\Vert _{H_{0}}^{2}+(m\circ {u_{x}})(t)>0, \ \ \ \lim _{\omega \rightarrow \infty }f\left( w\right) =-{\infty }, \end{aligned}$$we deduce that there exist exactly one \(\omega _*>0\) such that \(f(\omega _* )=0,\) i.e., \( \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)|_{\omega =\omega _*}=0.\)
-
(3)
Since \(\omega f(\omega )=\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)\), we obtain that \(\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)>0\) when \(\omega \in (0,\omega _*)\), \(\frac{\hbox {d}}{\hbox {d}\omega }J(\omega u)<0\) when \(\omega \in (\omega _*,\infty )\). Therefore, the conclusion of (3) holds.
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(4)
From \(I(\omega u)=\omega \frac{\hbox {d}}{\hbox {d}\omega }J(\omega u) \) and (3), we can immediately conclude that (4) holds.
\(\square \)
Proof
(Proof of Lemma 2.5.) From (2.6) and definitions of N and d, we know that \(J(u)>0\) for \(u\in N\) and \(d\ge 0\). Now, we will show that there exists a positive function \(u\in N\) and satisfies \(J(u)=d\). In fact, we choose \(\{{u_i}\}_{i=1}^{\infty }\subset N\) to be a minimizing sequence for J. It means that
Thus, we can see that \(\{{|u_i|}\}_{i=1}^{\infty }\subset N\) is also a minimizing sequence for J. Hence, without loss of generality, we suppose \(u_{i}>0\) a.e. in \( (0,\iota ).\) Since \(\{J({u_i})\}_{i=1}^{\infty } \) is bounded and \( I(u_{i})=0\), we obtain from (2.6) that \(\{{u_i}\}_{i=1}^{\infty }\) is bounded in \( H_{0} \)-norm. Here, we pick \(\nu >0 \) small enough such that \(H_{0} \hookrightarrow L_{x}^{r+\nu }\) (\(2<r+\nu <4\)) is compact. Thus, we get that there exists a function u and a subsequence of \(\{u_i\}_{i=1}^\infty \) denoted by \(\{u_{{i}}\}_{i=1}^\infty \) satisfying
Therefore, we get that \(u\ge 0 \) a.e. in \((0,\iota )\).
Next, applying dominated convergence theorem, it follows that
By the weak lower semi-continuity of \(\Vert \cdot \Vert _{H_{0}}\), we have
In view of (6.6)–(6.8) and definition of J(u), there appears the relation
Since \(\{u_{i}\}\subset N,\) then we have \(I(u_{i})=0\). Thus, the combination of (6.6)–(6.8) and definition of I(u), we get
When \(I(u)=0\), we discover from assumption (A1) that
where applying the fact \(y^{-\nu }\ln y\le (e\nu )^{-1}\) \((y,\nu >0)\) and Sobolev embedding inequality \(\Vert u_{{i}}\Vert _{r+\nu }\le B_{1}\Vert u_{{i}}\Vert _{H_{0}}.\) The above inequality implies that
which together (6.6) yields \(\int \limits _{0}^{\iota }x|u|^{r}\ln |u|\hbox {d}x\ge k\left( \frac{ke\nu }{B_1^{r+\nu }}\right) ^\frac{2}{r+\nu -2}>0\). Thus, we conclude that \(u\ne 0\).
If \(I(u)<0\), by the Lemma 2.2 (4), we know that there exist a \(\omega _*\) such that \(0<\omega _*<1\) and \(I(\omega _* u)=0.\) Hence, a series of calculation gives that
Considering \(\omega _{*}^{2}<1\), the above formula is impossible for case \(d>0.\) Hence, by (6.10), we get \(I(u)=0\), which means \(u\in N.\) Combining (6.9) and definition of d, we deduce that \(J(u)=d\). \(\square \)
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Di, H., Qiu, Y. Well-posedness, asymptotic stability and blow-up results for a nonlocal singular viscoelastic problem with logarithmic nonlinearity. Z. Angew. Math. Phys. 75, 34 (2024). https://doi.org/10.1007/s00033-023-02177-5
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DOI: https://doi.org/10.1007/s00033-023-02177-5
Keywords
- Nonlocal singular viscoelastic problem
- Logarithmic nonlinearity
- Well-posedness
- Blow-up
- Asymptotic stability